This invention relates generally to transmission and modulation systems for digital data, and more particularly, to a digital data modulation system wherein bandwidth efficiency and the data transmission rate are improved by demultiplexing the digital data into a plurality of data channels, wherein the data pulses are shaped by signals having preselected characterizing functions, illustratively sinusoidal and cosinusoidal functions which have a quadrature phase relationship.
Spectral congestion due to ever increasing demand for digital transmission calls for spectrally efficient modulation schemes. Spectrally efficient modulation can be loosely said to be the use of power to save bandwidth, much as coding may be said to be the use of bandwidth to save power. In other words the primary objective of a spectrally efficient modulation scheme is to maximize the bandwidth efficiency (b), defined as the ratio of data rate (R.sub.b) to channel bandwidth (W). Since a signal can not be both strictly duration-limited and strictly band-limited, there are two approaches in designing a spectrally efficient data transmission scheme. One is the band-limiting approach; the other is the time-limiting approach. In the former, a strictly bandlimited spectral shape is carefuly chosen for the data pulse so as to satisfy the Nyquist criterion of zero intersymbol interference (ISI). In the latter, the data pulse is designed to have a short duration and the definition of bandwidth is somewhat relative depending on the situation involved. The latter approach is followed herein.
Like bandwidth, power is also a costly resource in radio transmission. So another objective in designing a high rate data transmission scheme is to reduce the average energy per bit (E.sub.b) for achieving a specified bit error rate (BER). The bit error rate performance of two schemes are usually compared under the assumption of an ideal channel corrupted only by additive white Gaussian noise (AWGN). Suppose the two sided power spectral density of noise is N.sub.o /2. Then a standard parameter for comparing two modulation schemes is the energy efficiency (e); it is the ratio E.sub.b /N.sub.o required to achieve a BER of 10.sup.-5.
The energy efficiency mostly depends on the signal space geometry. The bandwidth efficiency depends on two factors; firstly the basic waveforms of the data shaping pulses and secondly the utilization of all possible signal dimensions available within the given transmission bandwidth. In data communication, the motion of increasing the rate of transmission by increasing the number of dimensions became prominent when people switched from binary phase shift keying (BPSK) to quadrature phase shift keying (QPSK). Modulation studies during the last fifteen years proposed several modifications of QPSK. Of these, offset quadrature phase shift keying (OQPSK) and minimum shift keying (MSK) have gained popularity because of their several attributes. So any new spectrally efficient scheme ought to be tested in the light of the spectral efficiencies of these two.
BPSK is an antipodal signalling scheme; it uses two opposite phases (0.degree. and 180.degree.) of the carrier to transmit binary +1 and -1. Thus BPSK signal space geometry is one dimensional. QPSK, on the other hand, can be considered as two BPSK systems in parallel; one with a sine carrier, the other with a cosine carrier of the same frequency. QPSK signal space is thus two dimensional. This increase in dimension without altering the transmission bandwidth increases the bandwidth efficiency by a factor of two. Spectral compactness is further enhanced in MSK by using a cosinusoidal data shaping pulse instead of the rectangular one of QPSK. Though MSK over QPSK use different data shaping pulses, their signal space geometries are the same. Both of them use a set of four biorthogonal signals. So the spectral compactness achieved in MSK and QPSK should be distinguished from the compactness achieved in QPSK over BPSK. In the former compactness comes from the shaping of the data pulse, while in the later it comes from increasing the dimension within the given transmission bandwidth.
To see the possibility of any further increase in dimension without increasing the transmission bandwidth substantially, one has to look into the time-bandwidth product. It is a mathematical truth that the space of signals essentially limited in time to an interval .tau. and in one sided bandwidth occupancy to W is essentially 2.tau.W-dimensional. Though this bound on dimension is true for the best choice of orthonormal set (prolate spheroidal wave functions [4]), yet it will justify the reasoning behind any search for higher dimensional signal sets to achieve higher bandwidth efficiency. In both QPSK and MSK, signal duration .tau. is 2T, where T is the bit interval in the incoming data stream. Suppose the channel is bandlimited to 1/2T on either side of the carrier, i.e. one sided bandwidth occupancy is W=1/T. With such a bandlimited channel a QPSK system will be able to transmit only ninety percent of its total power while an MSK system transmits ninety seven percent. The number of dimensions available within this bandwidth W=1/T is 2.tau.W=4. It is surprising that only two of them are utilized in QPSK and MSK. The remaining two are yet left to be played around with. So one can conceive of a modulation scheme with a bandwidth efficiency as much as twice that of QPSK or MSK. Since prolate spheroidal wave functions are not easy to implement, expectation of one hundred percent increase in bandwidth efficiency may be too much from a practical view point. Yet the extra two dimensions give some room for improving the bandwidth efficiency by increasing the dimensionality of the signal set at the cost of some extra bandwidth, if necessary.
First we briefly review some existing modulation schemes such as QPSK, OQPSK and MSK, all of which use two dimensional signal sets. Then we propose a new modulation scheme which uses the vertices of a hyper cube of dimension four as the signal space geometry. This proposed scheme makes use of two data shaping pulses and two carriers which are pairwise quadrature in phase; so it is named quadrature-quadrature phase shift keying or Q.sup.2 PSK. It is pointed out as a theorem that in the presence of AWGN any modulation scheme which utilizes the vertices of some hyper cube as signal space geometry has the same energy efficiency; this is true for any dimension of the hyper cube. As a consequence of the theorem, Q.sup.2 PSK has the same energy efficiency as that of MSK, QPSK or OQPSK; but for a given bandwidth the transmission rate of Q.sup.2 PSK is twice that of any one of the three other schemes. However, all these four schemes respond differently when they undergo bandlimiting. Considering ninety nine percent bandwidth of MSK as the definition of channel bandwidth, it is shown that E.sub.b /N.sub.o requirement for achieving a BER of 10.sup.6 is 11.2 db for bandlimited Q.sup.2 PSK and 9.6 db for MSK. Thus bandlimited Q.sup.2 PSK achieves twice bandwidth efficiency of MSK only at the expense of 1.6 db or forty five percent increase in the average bit energy. Like MSK, Q.sup.2 PSK has also self synchronization ability. Modulator-demodulator implementation and a scheme for synchronization will be described in detail in the last section.